Searcharxiv⌕ Search

arXiv subjects

Jesse Wolfson

Publications and source records attributed to Jesse Wolfson.

31 records · Page 2Linked to original sources
arXiv2019

Coincidences of homological densities, predicted by arithmetic

Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil's number field/ function field analogy to predict coincidences for limiting homological densities of various sequences $\mathcal{Z}^{(d_1,\ldots,d_m)}_n(X)$ of spaces of $0$-cycles on manifolds $X$. The main theorem in this paper is that these topological predictions, which seem strange from a purely topological viewpoint, are indeed true. The obstacle to proving such a theorem with current technology is how to deal with the combinatorial complexity of all possible "collisions" of points, this problem does not arise in the simplest (and classical) case $(m,n)=(1,2)$ of configuration spaces. To overcome this obstacle we develop a method that uses the Björner--Wachs theory of lexicographic shellability from algebraic combinatorics to study such problems. As a consequence we derive new homological stability theorems for broad classes of $0$-cycles on manifolds. Even in the classical case $(m,n)=(1,2)$ this gives a new, simplified proof of classical results, and also of recent theorems of Church and others.

math.AT↗
arXiv2019

Topology and arithmetic of resultants, I: spaces of rational maps

We consider the interplay of point counts, singular cohomology, étale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree $d$ rational curves in $\mathbb{P}^n$. We deduce as special cases algebro-geometric and arithmetic refinements of topological computations of Segal, Cohen--Cohen--Mann--Milgram, Vassiliev and others.

math.AG↗
arXiv2018

The $A_\infty$-structure of the index map

Let $F$ be a local field with residue field $k$. The classifying space of $GL_n(F)$ comes canonically equipped with a map to the delooping of the $K$-theory space of $k$. Passing to loop spaces, such a map abstractly encodes a homotopy coherently associative map of A-infinity-spaces $GL_n(F)\to K_k$. Using a generalized Waldhausen construction, we construct an explicit model built for the $A_\infty$-structure of this map, built from nested systems of lattices in $F^n$. More generally, we construct this model in the framework of Tate objects in exact categories, with finite dimensional vector spaces over local fields as a motivating example.

math.KT↗
arXiv2018

The Index Map in Algebraic K-Theory

For a ring $R$, we construct a universal $K_R$-torsor $\mathcal{T}_R\to K_{Tate(R)}$ on the $K$-theory space of Tate $R$-modules. This torsor is closely related to canonical central extensions of loop groups. Just like classical loop group theory has features of $K$-theory (e.g. determinant bundles, tame symbol cocycle for Kac-Moody extension), the $K$-theory torsor relates higher loop groups with higher $K$-theory. We study the classifying "index" map of this torsor in detail. We explain how it arises in analogy with the classical index map of Fredholm operators, and we relate the $K$-theory torsor to previously studied dimension and determinant torsors.

math.KT↗
arXiv2017

Geometric and analytic structures on the higher adèles

The adèles of a scheme have local components - these are topological higher local fields. The topology plays a large role since Yekutieli showed in 1992 that there can be an abundance of inequivalent topologies on a higher local field and no canonical way to pick one. Using the datum of a topology, one can isolate a special class of continuous endomorphisms. Quite differently, one can bypass topology entirely and single out special endomorphisms (global Beilinson-Tate operators) from the geometry of the scheme. Yekutieli's "Conjecture 0.12" proposes that these two notions agree. We prove this.

math.AG↗
arXiv2016

Hochschild coniveau spectral sequence and the Beilinson residue

We develop the Hochschild analogue of the coniveau spectral sequence and the Gersten complex. Since Hochschild homology does not have devissage or A^1-invariance, this is a little different from the K-theory story. In fact, the rows of our spectral sequence look a lot like the Cousin complexes in Hartshorne's 1966 "Residues & Duality". Note that these are for coherent cohomology. We prove that they agree by an 'HKR isomorphism with supports'. Using the close ties of Hochschild homology to Lie algebra homology, this gives residue maps in Lie homology, which we show to agree with those à la Tate-Beilinson.

math.KT↗
arXiv2015

Étale homological stability and arithmetic statistics

We contribute to the arithmetic/topology dictionary by relating asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over $\Cb$ in the example of configuration spaces of $n$ points in smooth varieties. To do this, we import the method of homological stability from the realm of topology into the theory of étale cohomology; in particular we give the first examples of stability of étale cohomology groups as Galois representations where the Galois actions are not already explicitly known. We then establish subexponential bounds on the growth of the unstable cohomology, and we apply this and étale homological stability to compute the large $n$ limits of various arithmetic statistics of configuration spaces of varieties over $\F_q$.

math.AG↗
arXiv2015

Descent for n-Bundles

Given a Lie group G, one constructs a principal G-bundle on a manifold X by taking a cover U of X, specifying a transition cocycle on the cover, and descending the trivialized bundle along the cover. We demonstrate the existence of an analogous construction for local n-bundles for general n. We establish analogues for simplicial Lie groups of Moore's results on simplicial groups; these imply that bundles for strict Lie n-groups arise from local n-bundles. Our construction leads to simple finite dimensional models of Lie 2-groups such as String(n).

math.AT↗
arXiv2015

Relative Tate Objects and Boundary Maps in the K-Theory of Coherent Sheaves

We investigate the properties of relative analogues of admissible Ind, Pro, and elementary Tate objects for pairs of exact categories, and give criteria for those categories to be abelian. A relative index map is introduced, and as an application we deduce a description for boundary morphisms in the K-theory of coherent sheaves on Noetherian schemes.

math.KT↗
arXiv2015

Topology and arithmetic of resultants, II: the resultant $=1$ hypersurface (with an appendix by C. Cazanave)

We consider the moduli space $\mathcal{R}_n$ of pairs of monic, degree $n$ polynomials whose resultant equals $1$. We relate the topology of these algebraic varieties to their geometry and arithmetic. In particular, we compute their étale cohomology, the associated eigenvalues of Frobenius, and the cardinality of their set of $\mathbb{F}_q$-points. When $q$ and $n$ are coprime, we show that the étale cohomology of $\mathcal{R}_{n/\bar{\mathbb{F}}_q}$ is pure, and of Tate type if and only if $q\equiv 1$ mod $n$. We also deduce the values of these invariants for the finite field counterparts of the moduli spaces $\mathcal{M}_n$ of $SU(2)$ monopoles of charge $n$ in $\mathbb{R}^3$, and the associated moduli space $X_n$ of strongly centered monopoles. An appendix by Cazanave gives an alternative and elementary computation of the point counts.

math.AG↗
arXiv2015

Operator ideals in Tate objects

Tate's central extension originates from 1968 and has since found many applications to curves. In the 80s Beilinson found an n-dimensional generalization: cubically decomposed algebras, based on ideals of bounded and discrete operators in ind-pro limits of vector spaces. Kato and Beilinson independently defined '(n-)Tate categories' whose objects are formal iterated ind-pro limits in general exact categories. We show that the endomorphism algebras of such objects often carry a cubically decomposed structure, and thus a (higher) Tate central extension. Even better, under very strong assumptions on the base category, the n-Tate category turns out to be just a category of projective modules over this type of algebra.

math.AG↗
arXiv2014

Tate Objects in Exact Categories (with appendix by Jan \vS\vtov\'ı\vcek and Jan Trlifaj)

We study elementary Tate objects in an exact category. We characterize the category of elementary Tate objects as the smallest sub-category of admissible Ind-Pro objects which contains the categories of admissible Ind-objects and admissible Pro-objects, and which is closed under extensions. We compare Beilinson's approach to Tate modules to Drinfeld's. We establish several properties of the Sato Grassmannian of an elementary Tate object in an idempotent complete exact category (e.g. it is a directed poset). We conclude with a brief treatment of n-Tate modules and n-dimensional adèles. An appendix due to J. \vS\vtov\'ı\vcek and J. Trlifaj identifies the category of flat Mittag-Leffler modules with the idempotent completion of the category of admissible Ind-objects in the category of finitely generated projective modules.

math.KT↗